Ranges and Midranges

Sequences, Reads, Alignments, and Ranges

Projections in Operator Ranges

If H is a Hilbert space, A is a positive bounded linear operator on H and S is a closed subspace of H, the relative position between S and A−1(S⊥) establishes a notion of compatibility. We show that the compatibility of (A,S) is equivalent to the existence of a convenient orthogonal projection in the operator range R(A1/2) with its canonical Hilbertian structure.

Stimulation ranges, usage ranges, and paresthesia mapping during occipital nerve stimulation.

Introduction.   Subcutaneous, occipital nerve stimulation has emerged as a potentially effective treatment modality for patients with refractory headache disorders. The purpose of this study was to document occipital stimulation characteristics in 10 patients status post implantation of an occipital nerve stimulator. Methods.   All possible electrode combinations were tested in each patient, an...

Weaker Axioms, More Ranges

In the family of many-valued modal languages proposed by M. Fitting in 1992, every modal language is based on an underlying Heyting algebra which provides the space of truth values. The lattice of truth values is explicitly represented in the language by a set of special constants and this allows for forming weak, generalized, many-valued analogs of all classical modal axioms. Weak axioms of th...

Patterns of species ranges, speciation, and extinction.

The exact nature of the relationship among species range sizes, speciation, and extinction events is not well understood. The factors that promote larger ranges, such as broad niche widths and high dispersal abilities, could increase the likelihood of encountering new habitats but also prevent local adaptation due to high gene flow. Similarly, low dispersal abilities or narrower niche widths co...